Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

120 CHAPTER 5 Energy Methods

where the symbols take their previous meanings and is the vertical deflection of any point on the
beam.Then,

∂C

∂Pf

=

∫L

0

dθ

∂M

∂Pf

− (^) T= 0 (5.14)
Asbefore
dθ=

M

EI

dz

but

M=Pfz+

wz^2

2

(Pf= 0 )

Hence,

∂M

∂Pf

=z

SubstitutinginEq.(5.14)fordθ,Mand∂M/∂Pf,andrememberingthatPf=0,wehave

(^) T=

∫L

0

wz^3

2 EI

dz

giving

(^) T=
wL^4
8 EI
It will be noted that here, unlike the method for the solution of the pin-jointed framework, the
fictitiousloadisappliedtotheloadedbeam.Thereis,however,noarithmeticaladvantagetobegained
bytheformerapproachalthoughtheresultwouldobviouslybethesame,sinceMwouldequalwz^2 / 2
and∂M/∂Pfwouldhavethevaluez.
Example 5.2
Calculate the vertical displacements of the quarter and the midspan points B and C of the simply
supportedbeamoflengthLandtheflexuralrigidityEIloaded,asshowninFig.5.7.
ThetotalcomplementaryenergyCofthesystemincludingthefictitiousloadsPB,fandPC,fis

C=

∫

L

∫M

0

dθdM−PB,f (^) B−PC,f (^) C−

∫L

0

wdz (i)

Hence,

∂C

∂PB,f

=

∫

L

dθ

∂M

∂PB,f

− (^) B=0(ii)